These non-proximal regularizers facilitate solving for \(\lbrace x_t\rbrace \)  and \(\lbrace z_t\rbrace \)  since \(r_{1:t}(x)\!=\!\sigma _{1:t}\Vert x\Vert ^2/2\)  involves only one quadratic term and can be represented in constant memory space, unlike \(r_{1:t}(x)\!=\!\sum _{i=1}^t \sigma _i\Vert x\!-x_i\Vert ^2/2\)  that expands with time. On the other hand, non-proximal updates yield looser bounds, cf. [1]}, and require a new saddle-point analysis. Interestingly, they do not worsen the growth of LLP2, but do prevent it from achieving \(\mathcal {R}_T=\!\mathcal {O}(1)\)  for perfect predictions.
